The octal numeral system (octal system), or base-8 number system, represents numeric values using eight symbols, usually 0 to 7, providing the following symbols in ascending order: 0 1 2 3 4 5 6 7
To understand how octal numbers are constructed, it is useful to consider the decimal numeral system, or base-10 number system, which is the number system most commonly used. Digits in the decimal system are organised into columns and multiplied by powers of ten. For example, the columns for 123 are:
| 102 | 101 | 100 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 |
| 1 | * | 102 | = | 1 | * | ( 1 * 10 * 10 ) | = | 1 | * | 100 | = | 100 |
| 2 | * | 101 | = | 2 | * | ( 1 * 10 ) | = | 2 | * | 10 | = | 20 |
| 3 | * | 100 | = | 3 | * | ( 1 ) | = | 3 | * | 1 | = | 3 |
|
123
|
Decimal number 123 is ( 1 * 102 ) + ( 2 * 101 ) + ( 3 * 100 ).
If more detail is required, refer to the explanation of term decimal.
The octal system works under the same principles as the decimal system, except that it operates in base-8 rather than base-10. Therefore, the columns for octal number 173 are:
| 82 | 81 | 80 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 7 | 3 |
| 1 | * | 82 | = | 1 | * | ( 1 * 8 * 8 ) | = | 1 | * | 64 | = | 64 |
| 7 | * | 81 | = | 7 | * | ( 1 * 8 ) | = | 7 | * | 8 | = | 56 |
| 3 | * | 80 | = | 3 | * | ( 1 ) | = | 3 | * | 1 | = | 3 |
|
123
|
Octal number 173 is ( 1 * 82 ) + ( 7 * 81 ) + ( 3 * 80 ) = 123.
Instead of using digits 0 to 9 as the decimal system does, the octal system uses digits 0 to 7 to represent numbers. To put a larger number than 7 in column 8n it is neccesary to multiply 8 * 8n, which gives 8n+1 and carry a column to the left. For example, putting 12 in the 80 column is impossible, so a 1 is put in the 81 column and an 2 in the 80 column, thus using two columns. Ten is 10 * 80 = 80( 8 + 2 ) = ( 1 * 81 ) + ( 2 * 80 ).
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